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Session Overview
MS 18c: Advances in high order methods for fluid dynamic
Wednesday, 14/July/2021:
2:00pm - 4:00pm

Session Chair: Henar Herrero
Session Chair: Jezabel Curbelo
Virtual location: Zoom 2

Session Abstract

On approaching turbulence and geophysical domains, different scales appear in

the problems (Aurnou et al, 2019). Small scales and large domains are difficult to

reach with spectral methods because they are ill conditioned so that it is not

possible to increase suitably the size of the expansions. A strategy to increase

the size of the mesh avoiding the increase of the expansion are domain

decomposition (DD) techniques (Quarteroni, 1999, 2002). Among them, the

alternating Schwarz methods respect the original computation (Blayo, 2016; Xu,

2005). Herrero et al. applied this methodology to a stationary natural convection

problem (Herrero, 2018). The method is expensive in time and unstable for some

values of the parameters. A way of stabilization and optimization consists of using

two levels of mesh in the alternating Schwarz method (Axelsson, 2019).

Geophysical problems are studied using direct numerical simulations (GL Kooij

et al. 2019, Curbelo et al. 2019). The study of instabilities requires the tracking of

bifurcations varying parameters. These methods are based on the observation

that varieties calculated by means of proper orthogonal decomposition (POD)

from solutions at some values of the parameters also contain solutions to different

values of the parameters. If the equations that govern the evolution of a system

are dissipative, the behavior of the system for long times is contained in a finite-

dimensional inertial variety, which is often of a low dimension (Foias, 1988).

Herrero et al. applied this type of methods for a Rayleigh-Bénard problem in a

rectangular domain with reduced basis (Herrero, 2013) and taking as snapshots

transient states of a temporal evolution towards a stationary solution or transient

states of Newton's iteration for the treatment of non-linearity for a fixed parameter

value, managing to reproduce the diagram of bifurcations with very low

computational cost (Pla, 2015). Gutierrez-Castillo et al. analyzed the use of POD

for temporal flow patterns in non-newtonian fluids.

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2:00pm - 2:30pm

Fourth-order finite volume scheme for the simulation of rotating magnetoconvection at low magnetic Reynolds numbers

Susanne Horn

Coventry University, United Kingdom

Turbulent rotating magnetoconvection is fundamental to the fluid processes occurring deep inside planets, including the generation of the geodynamo in Earth's liquid metal core. But planetary interiors are shielded from direct observation, thus, idealised, physics-driven models are essential for gaining a detailed understanding. The canonical system is Rayleigh-Bénard convection, i.e. a fluid layer heated from below and cooled from above, rotated around the vertical axis and subject to an external magnetic field. Here, I will consider a cylindrical numerical set-up, which represents a small subvolume in the polar region of a planet and which is particularly relevant because it can be realised experimentally.

The governing equations are the Navier-Stokes equations coupled to the temperature and induction equation. In the quasi-static limit of low magnetic Reynolds numbers, i.e. when the magnetic fluctuations adapt almost instantaneously to the slowly varying velocity field and are much smaller than the external magnetic field, the equations simplify significantly. Instead of the full induction equation, most of the numerical effort is reduced to the solution of a Poisson equation for the electrostatic potential, which in turn can be used to calculate the Lorentz force in a straightforward manner. Nonetheless, the problem is stiff and many general-purpose CFD approaches cannot provide simulations with reasonable cost and accuracy for high magnetic field strengths and strong thermal forcing. This is especially true for liquid metal flows. The specialised code goldfish outperforms such general-purpose methods in terms of efficiency and accuracy. The code is based on a consistent and conservative finite volume scheme that is fourth-order accurate in space and uses a hybrid explicit/semi-implicit time integration method. The cylindrical meshes are staggered, equidistant in the azimuthal direction, and non-equidistant in radial and vertical direction ensuring adequate resolution of the bulk and all boundary layers. Direct numerical simulations are performed focussing on the geophysically relevant magnetostrophic regime where Lorentz and Coriolis forces are in balance.

2:30pm - 3:00pm

HDG vs FV: Turbulent incompressible flows

Hashim Elzaabalawy1, Ganbo Deng1, Luís Eça2, Michel Visonneau1

1CNRS/Ecole Centrale de Nantes, France; 2Técnico Lisboa, Universidade de Lisboa, Portugal

Solving the Reynolds averaged Navier-Stokes equations using high-order methods is known to be challenging due to their high stiffness. Strategies to overcome this problem under the hybridizable discontinuous Galerkin framework are presented for the Wilcox 98, TNT, BSL, and SST k − ω models. Special treatment of ω near the wall to fit the high-order polynomial approximation is proposed. Moreover, scaling limiters are introduced to preserve the positivity as well as the high-order accuracy of the turbulence variables. The turbulence model is coupled with the pointwise divergence-free hybridizable discontinuous Galerkin solver for incompressible flows and solved implicitly. The method of manufactured solution is employed to assess the formulation and the treatment of each equation separately. Further, the formulation is tested on the 2D channel flow, the zero pressure gradient flat plate, and the NACA 0012 airfoil test cases. Preliminary results show significant improvements regarding the error magnitudes and number of iterations compared to finite volume-based solvers. Additionally, optimal convergence rates are obtained. Finally, the possibilities and limitations for the high-order methods in RANS simulations are discussed and compared with the finite volume approach.

3:00pm - 3:30pm

Iteratively partitioned, high-order, multirate time stepping for fluid-fluid interaction with flux conservation

Jeffrey Connors

University of Connecticut, United States of America

Application codes that include air and sea simulation components often employ a concurrent configuration, in the sense that the air and sea codes run simultaneously on separate groups of processors. The resolution of air-sea interaction requires the exchange of data between the concurrent modules. Amongst the many challenges this problem presents, one is that the different time scales inherent in the two systems motivate the development of multirate coupling algorithms. Specifically, we consider a configuration that would synchronize the modules over a common interval of simulated time, or "coupling window", by setting the time step for the sea to be the length of the window and subcycling the air module with a smaller time step. Mathematically, time stepping has been studied in this context before using proxy models of two coupled fluids, known as fluid-fluid coupling. For these problems, unconditional stability has only been proved for methods that have a low formal order of accuracy with respect to the size of the coupling window, or that lack desirable conservation properties. We will discuss another approach that uses iterations between the concurrent modules on a coupling window to enforce conservation, higher-order accuracy and stability simultaneously. The properties are illustrated with some computational examples. In particular, we demonstrate the efficacy on large windows where another recent approach is inaccurate or even unstable.

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